A287090 Expansion of Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

1, 1, 10, 46, 191, 740, 2912, 10941, 40345, 144703, 509693, 1761738, 5993434, 20076668, 66329914, 216307961, 696990583, 2220665661, 7000973556, 21853019072, 67575353580, 207111103623, 629440843762, 1897670845715, 5677604053474, 16863081962184, 49736388996376, 145714874857754

Offset: 0

Author

Ilya Gutkovskiy, May 19 2017

Keywords

• nonn

Comments

Euler transform of A000537.

Links

• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to arXiv version]
• M. Bernstein and N. J. A. Sloane, Some canonical sequences of integers, Linear Alg. Applications, 226-228 (1995), 57-72; erratum 320 (2000), 210. [Link to Lin. Alg. Applic. version together with omitted figures]
• N. J. A. Sloane, Transforms

Crossrefs

Cf. A000294, A000537, A023872, A279215.

Programs

Mathematica

nmax = 27; CoefficientList[Series[Product[1/(1 - x^k)^(k^2 (k + 1)^2/4), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: Product_{k>=1} 1/(1 - x^k)^(k^2*(k+1)^2/4).

a(n) ~ exp(-Zeta(3) / (16*Pi^2) + 741*Zeta(5) / (1600*Pi^4) - 250047*Zeta(5)^3 / (5*Pi^14) + 10207918728 * Zeta(5)^5 / (5*Pi^24) + Zeta'(-3)/2 + (-7*(7/2)^(1/6) * Pi / (3200 * 3^(2/3)) + 9261 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^2 / (40*Pi^9) - 22754277 * 3^(1/3) * (7/2)^(1/6) * Zeta(5)^4 / (2*Pi^19)) * n^(1/6) + (-21 * 3^(2/3) * (7/2)^(1/3) * Zeta(5) / (20*Pi^4) + 31752 * 6^(2/3) * 7^(1/3) * Zeta(5)^3/Pi^14) * n^(1/3) + (sqrt(7/2)*Pi/60 - 567*sqrt(14)*Zeta(5)^2 / Pi^9) * sqrt(n) + 9 * 3^(1/3) * (7/2)^(2/3) * Zeta(5) / Pi^4 * n^(2/3) + 2 * (2/7)^(1/6) * 3^(2/3) * Pi/5 * n^(5/6)) / (2^(1321/1440) * 3^(479/720) * 7^(119/1440) * n^(839/1440) * Pi^(1/240)). - Vaclav Kotesovec, Nov 09 2017